Potential energy term in Schrodinger's equation?

consider a 1-D problem where a charged particle travels along +x with a bound state energy.

the vertical potential energy axis is displaying information about the force that is parallel to the particle's axis of motion. is this correct?

so then this would be an accurate representation?

where blue arrows represent E-field and the direction of force felt by a (+) test particle.
$$\lim_{ε→0}$$

so if it's just the force component that is parallel to the direction of motion that determines whether the particle is in a bound state or scattering state, please confirm the following analysis is correct:

consider a charged ideal parallel plate capacitor:

(where the fields do not leak outside the plate area)

positive test particle enters orthogonally to the field. but because the acting force on the particle is orthogonal to its motion, the effective potential energy for determining bound vs scattering state, is zero.
but at the end, the particle has a velocity component parallel to the force, so it has a chance to be bound or scattered along this vertical axis.

I'm confused about what you're talking about. First of all, are you saying that the potential energy is zero everywhere except at two points? If the potential is finite at those points then they have no effect, quantum-mechanically. Or do you mean that the "force" is zero everywhere, except at two points?

Also, you start off saying that it's a one-dimensional problem, but then you talk about the direction of the particle's motion. If the problem is 1-dimensional, then there are only two directions possible: to the left, or to the right. (Or in the +x direction vs in the -x direction)

Are you getting your problem statement from some book? Should this be in the homework forum?

ultimately what i wanted to get around asking was imagine a potential barrier in R3.

the red and pink represent the paths of two separate charge particles (sorry i didn't mean to draw them as if they were actually crossing one another. the focus is supposed to be between the barrier and each particle considered separately).
each particle has the same kinetic energy. but see how one particle is incident to the barrier at an angle and the other particle is incident at a right angle?
i just wanted to ask if the transmission/ reflection coefficients are solely determined by the kinetic energy component that is orthogonal to the barrier.